Homomorphism from Z into a nonabelian group

[quasar987]

What would be an example of a (nontrivial) homomorphism from Z into a nonabelian group??

More specifically, I am looking for two homomorphisms f, g: Z-->W such that for some n,m in Z f(n)g(m) does not commute.

 

[quasar987]

Ah! I found out that the smallest nonabelian group is the diedral group of order 6, D_3. And from its multiplication table, we see that embedded in it is a copy of Z_2 and a copy of Z_3 that contain elements that do not commute.

 

[StatusX]

A homomorphism from Z into any group G is completely specified by picking f(1), because if f(1)=g, then f(n)=gⁿ. And you can pick f(1) to be any element of G that you want, basically because Z is a free group.

 

[quasar987]

Thanks for that remark StatusX.